Analytical Solution to Optimal Relocation of Satellite Formation Flying in Arbitrary Elliptic Orbits

Contents lists available atSciVerse ScienceDirect

Aerospace Science and Technology

www.elsevier.com/locate/aescte

Analytical solution to optimal relocation of satellite formation flying in arbitrary

elliptic orbits

Hancheol Cho, Sang-Young Park

∗

, Sung-Moon Yoo, Kyu-Hong Choi

Astrodynamics & Control Lab., Dept. of Astronomy, Yonsei University, Seoul 120-749, Republic of Korea

a r t i c l e

i n f o

a b s t r a c t

Article history:

Received 19 January 2008

Received in revised form 10 October 2010 Accepted 5 January 2012

Available online 9 January 2012

Keywords:

Satellite formation flying Fuel-optimal reconfiguration Elliptic orbits

Fourier series Low-level thrust

The current paper presents and examines a general analytical solution to the optimal reconfiguration problem of satellite formation flying in an arbitrary elliptic orbit. The proposed approach does not use any simplifying assumptions regarding the eccentricity of the reference orbit. For the fuel optimal reconfiguration problem, continuous and variable low-thrust accelerations can be represented by the Fourier series and summed into closed-form solutions. Initial and final boundary conditions are used to establish the constraints on the thrust functions. The analytical solution can be implicated by the Fourier coefficients that minimize propellant usage during the maneuver. This solution is found that compares favorably with numerical simulations. Also, this analytical solution is very useful for designing a reconfiguration controller for satellite formation flying in a general elliptic orbit.

©2012 Elsevier Masson SAS. All rights reserved.

1. Introduction

In the future, advanced space applications will utilize forma-tion flying technologies that involve multiple satellites. Therefore, satellite formation flying requires technology drawn from vari-ous research fields such as relative orbit determination, formation keeping, formation reconfiguration, relative attitude determination, relative attitude control, etc. Among these technologies, the cur-rent paper focuses on formation reconfiguration. To build a desired formation or to change a formation shape, it is necessary to re-locate satellites into the desired relative positions between satel-lites. The reconfiguration of satellites is achieved by optimizing the thrust accelerations required. There has already been a vari-ety of research dealing with the problem of minimum propellant transfers for satellite reconfigurations in formation flying. Most of solutions have been numerically obtained because this problem is highly nonlinear. To employ a distributed computational archi-tecture, a hybrid optimization algorithm is developed for satel-lite formation reconfiguration[15]. As well, using the calculus of variations approach, the optimal reconfiguration trajectories are numerically determined[7]. An algorithm for the reconfiguration problem is presented based upon Hamilton–Jacobi–Bellman opti-mality to generate a set of maneuvers to move from an initial stable formation to a final stable formation[3]. A reconfiguration problem about an Earth–Sun libration point is solved by use of an

*

Corresponding author. Tel.: +82 2 2123 5687, fax: +82 2 392 7680. E-mail addresses:narziss@yonsei.ac.kr(H. Cho),spark@galaxy.yonsei.ac.kr (S.-Y. Park),smyoo@galaxy.yonsei.ac.kr(S.-M. Yoo),khchoi@galaxy.yonsei.ac.kr (K.-H. Choi).

algorithm with generating functions to provide two impulsive ma-neuvers[4]. Finding the numerical solutions is somewhat difficult because the necessary and optimality conditions must be numeri-cally satisfied. However, analytic solutions would give insight into the feedback controller, and therefore would be easily exploited for formation flying, if they could be uncovered. For reconfigu-ration maneuvers of formation flying, an analytical two-impulse solution is proposed using Gauss’s variational equations[12]. This algorithm is based on the circular reference orbit described by the Hill–Clohessy–Wiltshire (HCW) equations. An analytic solution has been published for the formation relocation of a satellite using continuous and variable thrust acceleration in order to adopt low-thrust maneuvers [8]. These analytic solutions are very useful in their application to formation maintenance and relocation. How-ever, this analytic solution is limited to formation flying in only a circular or near-circular orbit because the HCW equations are used in[8]. The relative motions in satellite formation become a more realistic and complex problem when the non-zero eccentricity of reference orbit (i.e., a Chief satellite’s orbit) is considered. There-fore, the current paper extends the previous results in [8]to the satellite formation relocation problem in a general elliptic orbit. The proposed approach does not use any simplifying assumptions regarding the eccentricity of the reference orbit. In particular, the current paper provides the first presentation of the explicit closed-form solutions to relocation of closed-formation flying in an elliptic refer-ence orbit

(

e

=

0

)

. The analytical method developed in this paper yields closed-forms of accelerations, closed-forms of position and velocity vectors and closed-form of performance index, for any for-mation reconfiguration. The analytical solutions can be applied for spacecraft formation reconfigurations in highly elliptic orbit such 1270-9638/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

as the Magnetosphere Multiscale Mission (MMS) [16]. The analy-sis does not take account of orbital perturbations nor nonlinearity, because Tschauner–Hempel equations[11] are used in the current study.

In the control problem in the current paper, the relative dy-namics in an elliptic reference orbit are used. The true anomaly of the satellite is also used as an independent variable for con-venience. The out-of-orbital plane motion is decoupled from the in-plane motion, so it can be handled independently. Although the in-plane motion is much more complicated than the out-of-orbit plane motion, the procedure for deriving an in-plane solution is similar to that for the case of an out-of-plane solution. To derive the analytical solution to the optimal reconfiguration problem, the inhomogeneous solution and the particular solution should be ana-lytically formulated. Initial and final positions and velocities of the Deputy satellites are calculated using Tschauner–Hempel equations

[11] in order to establish the constraints on the thrust functions. These constraints can be incorporated into the performance in-dex by introducing Lagrange multipliers. The analytical solution is formed by the magnitude and direction of thrust accelerations as a function of the true anomaly. It is assumed that there are no restrictions on the thrust vector, and a transfer time is chosen as a specific value. The satellites are assumed to have low-level thrusters in three orthogonal directions which correspond to the radial, in-track and cross-track directions, respectively. Thrusters are fired during a significant fraction of an orbital period through-out the maneuver. Any thruster acceleration can be represented by the infinite Fourier series. With Parseval’s theorem, the Fourier series can be summed into a closed-form solution. Analytical op-timal solutions can be derived by extremizing the performance index with respect to all of the Fourier coefficients. Then, the solu-tion minimizes propellant usage for the reconfigurasolu-tion of satellite formation. Performing the analytical solutions, the satellites can generate an optimal reconfiguration trajectory. The analytic solu-tions are valid for an arbitrary elliptic orbit satisfying 0

<

e

<

1 as well as a circular orbit. The present paper describes thrust acceler-ations in closed-form for the optimal satellite relocation problem and the solution will be very useful for designing a controller for satellite formation flying in a general elliptic orbit.

2. Relative orbital dynamics in an elliptic orbit

In this section, we briefly show relative dynamics for an ellip-tic orbit. This also provides the necessary equations to be used. Since the satellites are moving in an elliptical orbit, we should use the Tschauner–Hempel equations instead of Hill’s equations. These equations, which were first derived by Tschauner and Hempel[11], are as follows:

⎡

⎣

xy

¨

¨

¨

z

⎤

⎦ = −

2

⎡

⎣

0

˙θ

− ˙θ

0 00 0 0 0

⎤

⎦

⎡

⎣

˙

x

˙

y

˙

z

⎤

⎦ −

⎡

⎣

− ˙θ

2 _{0 0} 0

− ˙θ

2 ₀ 0 0 0

⎤

⎦

⎡

⎣

xy z

⎤

⎦

−

⎡

⎣

0

¨θ

− ¨θ

0 00 0 0 0

⎤

⎦

⎡

⎣

xy z

⎤

⎦ +

ρ

(θ )

3

Γ

4

⎡

⎣

₋

2xy

−

z

⎤

⎦ +

⎡

⎣

TTxy Tz

⎤

⎦

(1)

where the x

(

t

)

axis lies in the radial direction, the y

(

t

)

axis is in the in-track direction, and the z

(

t

)

axis along the orbital angular momentum vector completes a right-handed system (seeFig. 1), while the dot (·) represents the differentiation with respect to time

(

t

)

. In addition,

θ (

t

)

and e refer to the true anomaly and the eccentricity of the Chief satellite, respectively.

ρ

(θ )

≡

1

+

e cos

θ

and

Γ

≡

L3/2

_/

_{G M are defined, where L}

₌

_R2

˙θ

_{is the magnitude} of the orbital angular momentum of the Chief satellite, G is the universal gravitational constant, and M is the mass of Earth. It is assumed that the thrust

[

Tx

(

t

),

Ty

(

t

),

Tz

(

t

)

]

T can be applied at

Fig. 1. The description of relative motion[14].

the desired directions during the maneuver. Changing the indepen-dent variable from time

(

t

)

to the true anomaly

(θ )

, Eq.(1)can be rewritten as:

⎡

⎢

⎢

⎣

x x y y

⎤

⎥

⎥

⎦



=

⎡

⎢

⎢

⎣

0 1 0 0 1

+

2

/

ρ

−

2

ρ



/

ρ

2

ρ



/

ρ

2 0 0 0 1

−

2

ρ



/

ρ

−

2 1

−

1

/

ρ

−

2

ρ



/

ρ

⎤

⎥

⎥

⎦

⎡

⎢

⎢

⎣

x x y y

⎤

⎥

⎥

⎦

+

Γ

4

ρ

4

⎡

⎢

⎢

⎣

0 0 1 0 0 0 0 1

⎤

⎥

⎥

⎦



Tx Ty

(2a)



z z



=



0 1

−

1

/

ρ

−

2

ρ



/

ρ



z z

+

Γ

4

ρ

4



0 1

Tz (2b)

where primer

(



)

represents differentiation with respect to the true anomaly, and thrust vector T

= [

Tx

(θ ),

Ty

(θ ),

Tz

(θ )

]

Tand state

vector

[

x

(θ ),

y

(θ ),

z

(θ )]

T _{are now described by the true anomaly.}

ρ, e, and

Γ

are the same as noted above, and

ρ



= −

e sin

θ

. The true anomaly

(θ )

is easily calculated from time using Ke-pler’s equation. The in-plane

(

x

(θ )

and y

(θ ))

motion and the out-of-plane

(

z

(θ ))

motion are decoupled, so we can deal with the problems separately. Now, for brevity, let’s consider the following transformation:

[˜

x

,

y

˜

,

˜

z

]

T

=

ω

1/2

_[

_x

_,

_y

_,

_z

_]

T ₍₃₎ u

(θ )

=

ux

(θ ),

uy

(θ ),

uz

(θ )

T

=

Tx

(θ ),

Ty

(θ ),

Tz

(θ )

T

/

ω

3/2 ₍₄₎ where

ω

≡ ˙θ

, the orbital rate of the Chief satellite. With the same procedure as that derived by Humi[5], Eqs.(2a) and (2b)become very simple:

⎡

⎢

⎢

⎣

˜

x

˜

x

˜

y

˜

y

⎤

⎥

⎥

⎦



=

⎡

⎢

⎢

⎣

0 1 0 0 3

/

ρ

0 0 2 0 0 0 1 0

−

2 0 0

⎤

⎥

⎥

⎦

⎡

⎢

⎢

⎣

˜

x

˜

x

˜

y

˜

y

⎤

⎥

⎥

⎦ +

⎡

⎢

⎢

⎣

0 0 1 0 0 0 0 1

⎤

⎥

⎥

⎦



ux uy

(5a)



˜

z

˜

z



=



0 1

−

1 0



˜

z

˜

z

+



0 1

uz (5b)

It is important to note that the actual position

(

x

,

y

,

z

)

and velocity

(˙

x

,

˙

y

,

z

˙

)

are related to the pseudo-position

(˜

x

,

˜

y

,

˜

z

)

and pseudo-velocity

(˜

x

,

y

˜



,

˜

z

)

as follows:

⎡

⎣

xy

(θ )

(θ )

z

(θ )

⎤

⎦ =

Γ

ρ

(θ )

⎡

⎣

˜

x

˜

y

(θ )

(θ )

˜

z

(θ )

⎤

⎦

(6a) d dt

⎡

⎣

xy

(θ )

(θ )

z

(θ )

⎤

⎦ =

e sin

θ

Γ

⎡

⎣

xy

˜

˜

(θ )

(θ )

˜

z

(θ )

⎤

⎦ +

ρ

(θ )

Γ

⎡

⎣

x

˜



_{(θ )}

˜

y

(θ )

˜

z

(θ )

⎤

⎦

(6b) or

⎡

⎣

xy

˜

˜

(θ )

(θ )

˜

z

(θ )

⎤

⎦ =

ρ

(θ )

Γ

⎡

⎣

xy

(θ )

(θ )

z

(θ )

⎤

⎦

(7a)

⎡

⎣

x

˜



_{(θ )}

˜

y

(θ )

˜

z

(θ )

⎤

⎦ = −

e sin

θ

Γ

⎡

⎣

xy

(θ )

(θ )

z

(θ )

⎤

⎦ +

Γ

ρ

(θ )

d dt

⎡

⎣

xy

(θ )

(θ )

z

(θ )

⎤

⎦

(7b)

3. Solutions to relative orbital dynamics in an elliptic orbit

In this section, we derive the solutions to relative orbital dy-namics in an elliptic orbit. Eqs. (5a) and (5b)are key equations from which we start. Because the

˜

z motion is less difficult to deal with, we will consider the out-of-plane maneuvers first.

3.1. Solution to out-of-plane maneuvers Eq.(5b)is of the form:

Z

=

AzZ

+

Bzuz

As is well known, it has the following solution:



˜

z

(θ )

˜

z

(θ )

= Φ

z

(θ )Φ

−1z

(θ

0

)



˜

z

(θ

0

)

˜

z

(θ

0

)

+ Φ

z

(θ )

θ



θ0

Φ

−_z1

(

τ

)



0 1

uz

(

τ

)

d

τ

(8)

where

Φ

z

(θ )

is a fundamental matrix associated with Az

=

0 1 −1 0



, and

Φ

z

(θ )Φ

−z1

(θ

0

)

is the state transition matrix associated with Az

(θ )

.

θ

0 is the true anomaly when the thruster starts to fire, and

τ

is used as an integration variable. The first term on the right in Eq.(8)is a homogeneous solution, while the second is a particular solution which contains the thrust uz. It is straightforward to show

that

Φ

z

(θ )

=



cos

θ

sin

θ

−

sin

θ

cos

θ

(9a)

Φ

−1_z

(θ )

=



cos

θ

−

sin

θ

sin

θ

cos

θ

(9b)

Thus, the homogeneous solution to Eq.(5b)is

ˆ˜

z

(θ )

ˆ˜

z

(θ )



=



cos

(θ

− θ0

)

sin

(θ

− θ0

)

−

sin

(θ

− θ0

)

cos

(θ

− θ0

)



˜

z

(θ

0

)

˜

z

(θ

0

)

(10a) or



_ˆ

z

(θ )

˙ˆ

z

(θ )

=

_cos(θ−θ 0)+e cosθ 1+e cosθ Γ 2 sin(θ−θ0) (1+e cosθ )(1+e cosθ0) − 1

Γ2[sin(θ− θ0)+e(sinθ−sinθ0)]

cos(θ−θ0)+e cosθ0 1+e cosθ0



×



z

(θ

0

)

˙

z

(θ

0

)

(10b)

where we use the carat to denote its state in the absence of thrust; be careful about the difference between the actual state

(

z

)

and pseudo-state

(˜

z

)

. z

˜

(θ

0

)

and

˜

z

(θ

0

)

at

θ

0 must be calculated from the actual values z

(θ

0

)

andz

˙

(θ

0

)

using Eqs.(7a) and (7b). Inserting e

=

0 into Eq.(10b)yields only the homogeneous solution of Hill’s equations.

The particular solution to Eq.(5b)is

˜

zp

(θ )

=

sin

θ

θ



θ0 uz

(

τ

)

cos

τ

d

τ

−

cos

θ

θ



θ0 uz

(

τ

)

sin

τ

d

τ

= Γ

3 θ



θ0 sin

(θ

−

τ

)

ρ

(

τ

)

3 Tz

(

τ

)

d

τ

(11a)

˜

z_p

(θ )

=

cos

θ

θ



θ0 uz

(

τ

)

cos

τ

d

τ

+

sin

θ

θ



θ0 uz

(

τ

)

sin

τ

d

τ

= Γ

3 θ



θ0 cos

(θ

−

τ

)

ρ

(

τ

)

3 Tz

(

τ

)

d

τ

(11b)

where the following identity is used:

uz

(θ )

=

Tz

(θ )

ω

3/2

=

Γ

3

ρ

(θ )

3Tz

(θ )

In summary, the z thruster fires at

θ

0 and the satellite sweeps out

ϑ

during which the thruster fires continuously at a variable thrust magnitude; after this, it is located at

θ

= θ

0

+ ϑ

and its position and velocity can be obtained by adding the homogeneous solution to the particular solution:



˜

z

(θ )

˜

z

(θ )

=

ˆ˜

z

(θ )

ˆ˜

z

(θ )



+



˜

zp

(θ )

˜

z_p

(θ )

or



z

(θ )

˙

z

(θ )

=



_ˆ

z

(θ )

˙ˆ

z

(θ )

+



zp

(θ )

˙

zp

(θ )

When the Chief satellite arrives at a final true anomaly

(θ

F

)

, the

thruster is turned off and the Deputy satellite is in the desired rel-ative position. That is,z

˜

(θ

F

)

and

˜

z

(θ

F

)

(or z

(θ

F

)

andz

˙

(θ

F

)

) are our

predefined values, which give the constraints on the thrust func-tion. Of course, they must be transformed from the actual position and velocity by Eqs.(7a) and (7b). Then,



˜

zp

(θ

F

)

˜

z_p

(θ

F

)

=



˜

z

(θ

F

)

˜

z

(θ

F

)

−

ˆ˜

z

(θ

F

)

ˆ˜

z

(θ

F

)



where

ˆ˜

z

(θ

F

)

and z

ˆ˜



(θ

F

)

are the position and velocity at

θ

F if the

thrust has not been applied. We must find where the thrust uz

(θ )

meets z

˜

p

(θ

F

)

andz

˜

p

(θ

F

)

, as given in the above equation, so this

equation can be thought of as representing the boundary condi-tions.

3.2. Solution to in-plane maneuvers

In-plane motion seems to be somewhat cumbersome because x and y are coupled. This is because relative motion is described in a noninertial frame. However, the analysis is parallel to the out-of-plane case. The relative equation for in-out-of-plane motion (Eq.(5a)) is

Ξ



=

Axy

Ξ

+

BxyUxy

This equation has the following solution:

⎡

⎢

⎢

⎣

˜

x

(θ )

˜

x

(θ )

˜

y

(θ )

˜

y

(θ )

⎤

⎥

⎥

⎦ = Φ

xy

(θ )Φ

−1xy

(θ

0

)

⎡

⎢

⎢

⎣

˜

x

(θ

0

)

˜

x

(θ

0

)

˜

y

(θ

0

)

˜

y

(θ

0

)

⎤

⎥

⎥

⎦

+ Φ

xy

(θ )

θ



θ0

Φ

−1_xy

(

τ

)

⎡

⎢

⎢

⎣

0 0 1 0 0 0 0 1

⎤

⎥

⎥

⎦



ux

(

τ

)

uy

(

τ

)

d

τ

(12)

As in the out-of-plane case, the first term on the right is a ho-mogeneous solution which enables the Deputy satellite to follow its own specific trajectory and the second is a particular solution with which the Deputy can be forced to arrive at the desired state. The fundamental matrix

(Φ

xy

)

associated with Axyis that given by Yamanaka and Ankersen[14]:

Φ

xy

(θ )

=

⎡

⎢

⎢

⎢

⎣

0

−

s

−

c 3esK

−

2 0

−

s

−

c 3e

(

sK

+

s

/

ρ

2

₎

1

−

c

(

1

+

1

/

ρ

)

s

(

1

+

1

/

ρ

)

3

ρ

2K 0 2s 2c

−

e 3

(

1

−

2esK

)

⎤

⎥

⎥

⎥

⎦

(13a)

Φ

−_xy1

(θ

0

)

=

1 1

−

e2

⎡

⎢

⎣

−3e(s/ρ)(1+1/ρ) ec−2 1−e2 _{−es(1+1/ρ)} 3(s/ρ)(1+e2_{/ρ) 2e−c 0 s(1+1/ρ)} 3(c/ρ+e) s 0 c(1+1/ρ)+e 1−e2_{−3ρ −es 0 −ρ}2

⎤

⎥

⎦

θ=θ0 (13b)

where s

=

ρ

sin

θ

, c

=

ρ

cos

θ

, s

=

cos

θ

+

e cos 2

θ

, c

= −(

sin

θ

+

e sin 2

θ )

, and K

(θ )

=

1

Γ

2

(

t

−

t0

)

=

θ



θ0 1

ρ

(

τ

)

2d

τ

(14)

It is convenient to use K

(θ )

because it is directly obtained by observing the amount of time passed. Thus, the homogeneous so-lution to Eq.(5a)is

⎡

⎢

⎢

⎢

⎣

ˆ˜

x

(θ )

ˆ˜

x

(θ )

ˆ˜

y

(θ )

ˆ˜

y

(θ )

⎤

⎥

⎥

⎥

⎦

= Φ

xy

(θ )Φ

−1xy

(θ

0

)

⎡

⎢

⎢

⎣

˜

x

(θ

0

)

˜

x

(θ

0

)

˜

y

(θ

0

)

˜

y

(θ

0

)

⎤

⎥

⎥

⎦

(15a) or

⎡

⎢

⎢

⎣

ˆ

x

(θ )

˙ˆ

x

(θ )

ˆ

y

(θ )

˙ˆ

y

(θ )

⎤

⎥

⎥

⎦

=

⎡

⎢

⎢

⎣

Γ /

ρ

(θ )

0 0 0 e sin

θ/Γ

ρ

(θ )/Γ

0 0 0 0

Γ /

ρ

(θ )

0 0 0 e sin

θ/Γ

ρ

(θ )/Γ

⎤

⎥

⎥

⎦

× Φ

xy

(θ )Φ

−xy1

(θ

0

)

×

⎡

⎢

⎢

⎣

ρ

(θ

0

)/Γ

0 0 0

−

e sin

θ

0

/Γ

Γ /

ρ

(θ

0

)

0 0 0 0

ρ

(θ

0

)/Γ

0 0 0

−

e sin

θ

0

/Γ

Γ /

ρ

(θ

0

)

⎤

⎥

⎥

⎦

×

⎡

⎢

⎢

⎣

x

(θ

0

)

˙

x

(θ

0

)

y

(θ

0

)

˙

y

(θ

0

)

⎤

⎥

⎥

⎦

(15b)

where the carat is again used to represent the state of the Deputy satellite if the thrust has not been applied.

Next, the particular solution to Eq.(5a) must be found, which requires the inverse of

Φ

xy

(θ )

. Let

Φ

−xy1

(θ )

take the following form:

Φ

−1_xy

(θ )

=

⎡

⎢

⎢

⎣

P11 P12 P13 P14 P21 P22 P23 P24 P31 P32 P33 P34 P41 P42 P43 P44

⎤

⎥

⎥

⎦

where the components Pi j (i

,

j

=

1

,

2

,

3

,

4) are given inAppendix A.

If we define Q

(θ )

≡

⎡

⎢

⎢

⎣

Q2

(θ )

Q3

(θ )

Q4

(θ )

Q5

(θ )

⎤

⎥

⎥

⎦ ≡

θ



θ0

Φ

−_xy1

(

τ

)

⎡

⎢

⎢

⎣

0 0 1 0 0 0 0 1

⎤

⎥

⎥

⎦



ux

(

τ

)

uy

(

τ

)

d

τ

then the particular solution will be

⎡

⎢

⎢

⎣

˜

xp

(θ )

˜

x_p

(θ )

˜

yp

(θ )

˜

y_p

(θ )

⎤

⎥

⎥

⎦ = Φ

xy

(θ )

θ



θ0

Φ

−_xy1

(

τ

)

⎡

⎢

⎢

⎣

0 0 1 0 0 0 0 1

⎤

⎥

⎥

⎦



ux

(

τ

)

uy

(

τ

)

d

τ

=

⎡

⎢

⎢

⎣

0

−

s

−

c 3esK

−

2 0

−

s

−

c 3e

(

sK

+

s

/

ρ

2

₎

1

−

c

(

1

+

1

/

ρ

)

s

(

1

+

1

/

ρ

)

3

ρ

2_K 0 2s 2c

−

e 3

(

1

−

2esK

)

⎤

⎥

⎥

⎦

×

⎡

⎢

⎢

⎣

Q2

(θ )

Q3

(θ )

Q4

(θ )

Q5

(θ )

⎤

⎥

⎥

⎦

(16) where

⎡

⎢

⎢

⎣

Q2

(θ )

Q3

(θ )

Q4

(θ )

Q5

(θ )

⎤

⎥

⎥

⎦ =

⎡

⎢

⎢

⎣

S

(

P12ux

)

+

S

(

P14uy

)

S

(

P22ux

)

+

S

(

P24uy

)

S

(

P32ux

)

+

S

(

P34uy

)

S

(

P42ux

)

+

S

(

P44uy

)

⎤

⎥

⎥

⎦

and S

(·)

which means an integration is defined as

S

(

·) =

S

·(θ)



=

θ



θ0

·(

τ

)

d

τ

After thrust is applied during

θ

, the satellite will reach a pre-defined state

[˜

x

(θ

F

),

x

˜



(θ

F

),

y

˜

(θ

F

),

y

˜



(θ

F

)

]

T. This places constraints

on ux

(θ )

and uy

(θ )

, creating the following relationships (from

Eq.(16)):

⎡

⎢

⎢

⎣

Q2

(θ

F

)

Q3

(θ

F

)

Q4

(θ

F

)

Q5

(θ

F

)

⎤

⎥

⎥

⎦ =

⎡

⎢

⎢

⎣

P11 P12 P13 P14 P21 P22 P23 P24 P31 P32 P33 P34 P41 P42 P43 P44

⎤

⎥

⎥

⎦

θ=θF

⎡

⎢

⎢

⎣

˜

xp

(θ

F

)

˜

x_p

(θ

F

)

˜

yp

(θ

F

)

˜

y_p

(θ

F

)

⎤

⎥

⎥

⎦

That is, we should find the ux

(θ )

and uy

(θ )

which satisfy the

above equation at

θ

= θ

F. Pi j (i

,

j

=

1

,

2

,

3

,

4) are given in Ap-pendix Aand the desired state

[˜

x

(θ

F

),

˜

x

(θ

F

),

y

˜

(θ

F

),

˜

y

(θ

F

)]

T is set

at the actual desired position

(

x

,

y

,

z

)

and velocity

(˙

x

,

y

˙

,

˙

z

)

using Eqs.(7a) and (7b).

In summary, when the Chief has

θ

(

θ

0

 θ  θ

F), the Deputy’s

position and velocity in the x– y plane will be

⎡

⎢

⎢

⎣

˜

x

(θ )

˜

x

(θ )

˜

y

(θ )

˜

y

(θ )

⎤

⎥

⎥

⎦ =

⎡

⎢

⎢

⎢

⎣

ˆ˜

x

(θ )

ˆ˜

x

(θ )

ˆ˜

y

(θ )

ˆ˜

y

(θ )

⎤

⎥

⎥

⎥

⎦

+

⎡

⎢

⎢

⎣

˜

xp

(θ )

˜

x_p

(θ )

˜

yp

(θ )

˜

y_p

(θ )

⎤

⎥

⎥

⎦

or

⎡

⎢

⎢

⎣

x

(θ )

˙

x

(θ )

y

(θ )

˙

y

(θ )

⎤

⎥

⎥

⎦ =

⎡

⎢

⎢

⎣

ˆ

x

(θ )

˙ˆ

x

(θ )

ˆ

y

(θ )

˙ˆ

y

(θ )

⎤

⎥

⎥

⎦ +

⎡

⎢

⎢

⎣

xp

(θ )

˙

xp

(θ )

yp

(θ )

˙

yp

(θ )

⎤

⎥

⎥

⎦

The above equation could be rewritten as

⎡

⎢

⎢

⎣

˜

x

(θ )

˜

x

(θ )

˜

y

(θ )

˜

y

(θ )

⎤

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎣

ˆ˜

x

(θ )

ˆ˜

x

(θ )

ˆ˜

y

(θ )

ˆ˜

y

(θ )

⎤

⎥

⎥

⎥

⎦

+

⎡

⎢

⎢

⎣

0

−

s

−

c 3esK

−

2 0

−

s

−

c 3e

(

sK

+

s

/

ρ

2

₎

1

−

c

(

1

+

1

/

ρ

)

s

(

1

+

1

/

ρ

)

3

ρ

2_K 0 2s 2c

−

e 3

(

1

−

2esK

)

⎤

⎥

⎥

⎦

×

⎡

⎢

⎢

⎣

S

(

P12ux

)

+

S

(

P14uy

)

S

(

P22ux

)

+

S

(

P24uy

)

S

(

P32ux

)

+

S

(

P34uy

)

S

(

P42ux

)

+

S

(

P44uy

)

⎤

⎥

⎥

⎦

In the end, the Chief arrives at

θ

F, then the x and y thrusters of

the Deputy are turned off and the Deputy satisfies the constraints mentioned earlier:

⎡

⎢

⎢

⎣

Q2

(θ

F

)

Q3

(θ

F

)

Q4

(θ

F

)

Q5

(θ

F

)

⎤

⎥

⎥

⎦ =

⎡

⎢

⎢

⎣

S

(

P12ux

)

+

S

(

P14uy

)

S

(

P22ux

)

+

S

(

P24uy

)

S

(

P32ux

)

+

S

(

P34uy

)

S

(

P42ux

)

+

S

(

P44uy

)

⎤

⎥

⎥

⎦

θ=θF

=

⎡

⎢

⎢

⎣

P11 P12 P13 P14 P21 P22 P23 P24 P31 P32 P33 P34 P41 P42 P43 P44

⎤

⎥

⎥

⎦

θ=θF

⎡

⎢

⎢

⎣

˜

xp

(θ

F

)

˜

x_p

(θ

F

)

˜

yp

(θ

F

)

˜

y_p

(θ

F

)

⎤

⎥

⎥

⎦

=

⎡

⎢

⎢

⎣

P11 P12 P13 P14 P21 P22 P23 P24 P31 P32 P33 P34 P41 P42 P43 P44

⎤

⎥

⎥

⎦

θ=θF

⎡

⎢

⎢

⎢

⎣

˜

x

(θ

F

)

− ˆ˜

x

(θ

F

)

˜

x

(θ

F

)

− ˆ˜

x 

(θ

F

)

˜

y

(θ

F

)

− ˆ˜

y

(θ

F

)

˜

y

(θ

F

)

− ˆ˜

y 

(θ

F

)

⎤

⎥

⎥

⎥

⎦

Of course, the states above have pseudo-values, so they must be evaluated from the actual values by Eqs.(7a) and (7b).

4. Thrust accelerations in a Fourier series

Our objective is to relocate the Deputy to the desired position relative to the Chief while minimizing fuel consumption. In gen-eral, the following performance index is used for fuel-optimality

J

=

tF



t0 TTRT dt

=

θF



θ0 TTRTdt d

θ

d

θ

= Γ

2 θF



θ0 TT_RT

ρ

2

_{(θ )}

d

θ

(17)

where T

= [

Tx Ty Tz

]

Tis a thrust acceleration vector of the Deputy

and the matrix R is a 3 by 3 weight matrix. In this paper, however, the following performance index is employed:

J

=

θF



θ0 TT

(

τ

)

T

(

τ

)

d

τ

=

θF



θ0 T

(

τ

)

2d

τ

(18)

where

τ

is used as an integration variable, T

(

τ

)

2

=

Tx

(

τ

)

2

+

Ty

(

τ

)

2

+

Tz

(

τ

)

2, and the low levels of thrusters are operated for

the Chief satellite’s

θ

0

 θ  θ

F. Although the control law

devel-oped from Eq.(18)is not strictly fuel-optimal and the effect of the denominator

ρ

2

_{(θ )}

_{in Eq. (17) is not negligible for moderate or} high eccentricities, setting a performance index as Eq.(18)enables a complete analytical approach as will be seen. Also, the effect of

ρ

(θ )

can be mitigated by choosing an appropriate gain matrix R and the use of Eq.(18)naturally penalizes the control effort near perigee[10,9]. Because the out-of-plane motion is decoupled from the plane motion, we can define the next two performance in-dices from Eq.(18):

Jz

=

θF



θ0 Tz

(

τ

)

2d

τ

(19a) and Jxy

=

θF



θ0

Tx

(

τ

)

2

+

Ty

(

τ

)

2



d

τ

(19b)

We must also incorporate the constraints formulated in the pre-vious section and find the thrust functions in terms of the true anomaly. The question is how we can represent general thrust functions. Using a Fourier series can yield an appropriate answer. While thrust functions may be discontinuous or impossible to dif-ferentiate, generating a description of thrust function by a Fourier series can solve this troublesome problem. Since it satisfies Dirich-let conditions, that is, it has a finite number of finite discontinu-ities and has a finite number of extrema, any thrust function can be mathematically represented in a Fourier series that converges to the function at continuous points and the mean of the posi-tive and negaposi-tive limits at points of discontinuity. In brief, we can think of each thrust function as a Fourier series with the period

θ

= θ

F

− θ

0. Then, the thrust acceleration in a Fourier series with domain

θ

0

 θ  θ

F can be obtained. Therefore, the performance

indices can be represented by Fourier coefficients that satisfy the preceding constraints.

Thrust functions in a Fourier series are obtained as follows:

Tx

(θ )

=

ax0 2

+

∞



n=1



axncos 2n

π

θ

θ

+

bxnsin 2n

π

θ

θ



(20a) Ty

(θ )

=

ay0 2

+

∞



n=1



ayncos 2n

π

θ

θ

+

bynsin 2n

π

θ

θ



(20b) Tz

(θ )

=

az0 2

+

∞



n=1



azncos 2n

π

θ

θ

+

bznsin 2n

π

θ

θ



(20c) where ax0

=

2

θ

θF



θ0 Tx

(θ )

d

θ,

axn

=

2

θ

θF



θ0 Tx

(θ )

cos 2n

π

θ

θ

d

θ

bxn

=

2

θ

θF



θ0 Tx

(θ )

sin 2n

π

θ

θ

d

θ

ay0

=

2

θ

θF



θ0 Ty

(θ )

d

θ,

ayn

=

2

θ

θF



θ0 Ty

(θ )

cos 2n

π

θ

θ

d

θ

byn

=

2

θ

θF



θ0 Ty

(θ )

sin 2n

π

θ

θ

d

θ

az0

=

2

θ

θF



θ0 Tz

(θ )

d

θ,

azn

=

2

θ

θF



θ0 Tz

(θ )

cos 2n

π

θ

θ

d

θ

bzn

=

2

θ

θF



θ0 Tz

(θ )

sin 2n

π

θ

θ

d

θ

If Parseval’s theorem [2], which represents the relationship be-tween the average of the square of T

(θ )

and the Fourier coeffi-cients, is used, the performance index for the z thrust function can be written as Jz

=

θ

2

a2_z0 2

+

∞



n=1



a2_zn

+

b2_zn





(21a)

and the performance index for the coupled x and y thrust acceler-ations is Jxy

=

θ

2

a2_x0 2

+

∞



n=1



a2_xn

+

b2_xn





+

θ

2

a2 y0 2

+

∞



n=1



a2_yn

+

b2_yn





(21b)

Now, we must find those Fourier coefficients

(

ax0

,

axn

,

bxn

,

ay0

,

ayn

,

byn

)

which minimize the performance indices Jxy and Jz. In

doing so, we must not forget to incorporate boundary constraints. For this, it is best to represent the boundary constraints in terms of Fourier coefficients. We find those coefficients which yield the op-timal thrust accelerations by minimizing the performance indices with respect to the coefficients. Let us consider the out-of-plane case first, and then the in-plane case.

4.1. Out-of-plane thrust functions

For the out-of-plane thrust function, there exist constraints on

˜

z

(θ

F

)

and

˜

z

(θ

F

)

because we wish to set the Deputy to a desired

state. The particular solutions which indicate the thrust neces-sary to put the Deputy into the desired state can be thought of as boundary conditions. Here, rather than the original constraints,

˜

z

(θ

F

)

andz

˜



(θ

F

)

, new transformed constraints I0 and I1 are intro-duced for a simpler calculation. This is just a coordinate transfor-mation:



I0 I1

=



sin

θ

F cos

θ

F

−

cos

θ

F sin

θ

F



˜

zp

(θ

F

)

˜

z_p

(θ

F

)

=

⎡

⎣

Γ

3



θF θ0 cosτ ρ(τ)3Tz

(

τ

)

d

τ

Γ

3



θF θ0 sinτ ρ(τ)3Tz

(

τ

)

d

τ

⎤

⎦

(22)

Substituting Eq.(20c)into Eq.(22), we obtain

I0

=

fz0az0 2

+

∞



n=1 fza

(

n

)

azn

+

∞



n=1 fzb

(

n

)

bzn (23a) I1

=

gz0az0 2

+

∞



n=1 gza

(

n

)

azn

+

∞



n=1 gzb

(

n

)

bzn (23b) where fz0

= Γ

3 θF



θ0 cos

τ

(

1

+

e cos

τ

)

3d

τ

fza

(

n

)

= Γ

3 θF



θ0 cos

τ

(

1

+

e cos

τ

)

3cos 2n

π

θ

τ

d

τ

fzb

(

n

)

= Γ

3 θF



θ0 cos

τ

(

1

+

e cos

τ

)

3sin 2n

π

θ

τ

d

τ

(24a) gz0

= Γ

3 θF



θ0 sin

τ

(

1

+

e cos

τ

)

3d

τ

gza

(

n

)

= Γ

3 θF



θ0 sin

τ

(

1

+

e cos

τ

)

3cos 2n

π

θ

τ

d

τ

gzb

(

n

)

= Γ

3 θF



θ0 sin

τ

(

1

+

e cos

τ

)

3sin 2n

π

θ

τ

d

τ

(24b)

where

θ

= θ

F

− θ

0. It is noted that Eqs.(23a) and (23b) repre-sent the constraints in terms of Fourier coefficients. Incorporating the constraints (Eqs. (23a) and (23b)) using Lagrange multipliers

λ

0 and

λ

1, the augmented performance index Jz,aug obtains the

following: Jz,aug

=

θ

2

a2_z0 2

+

∞



n=1 a2_zn

+

∞



n=1 b2_zn



+ λ0

I0

−

fz0az0 2

−

∞



n=1 fza

(

n

)

azn

−

∞



n=1 fzb

(

n

)

bzn



+ λ1

I1

−

gz0az0 2

−

∞



n=1 gza

(

n

)

azn

−

∞



n=1 gzb

(

n

)

bzn



(25)

Then, partially differentiating Eq.(25)with respect to the respec-tive Fourier coefficients az0, azn, bzn, and setting the results equal

to zero, the coefficients for the optimal maneuver are obtained as follows to minimize the performance index:

az0

=

1

θ

[λ0

fz0

+ λ1

gz0

]

azn

(

n

)

=

1

θ

λ

0fzn

(

n

)

+ λ1

gzn

(

n

)



bzn

(

n

)

=

1

θ

λ

0fzb

(

n

)

+ λ1

gzb

(

n

)



(26)

Substituting Eq. (26) into Eqs.(23a) and (23b), I0 and I1 are rewritten in terms of

λ

0 and

λ

1:



I0 I1

=



p0 p1 p1 q1



λ

0

λ

1

(27) where p0

=

f2 z0 2

θ

+

1

θ

∞



n=1

fza

(

n

)

2

+

fzb

(

n

)

2



p1

=

fz0gz0 2

θ

+

1

θ

∞



n=1

fza

(

n

)

gza

(

n

)

+

fzb

(

n

)

gzb

(

n

)



q1

=

g2_z0 2

θ

+

1

θ

∞



n=1

gza

(

n

)

2

+

gzb

(

n

)

2



p0, p1, and q1 contain the infinite series. However, by Parseval’s theorem, all of these infinite sums converge at constant values:

p0

=

Γ

6 2 θF



θ0 cos2

_τ

(

1

+

e cos

τ

)

6d

τ

=

Γ

6 2

(

1

−

e2

₎

11/2



−

1 80e 3_{sin 5E}

₊

1 32e 2

₍

_2e2

₊

₃

₎

_{sin 4E}

−

1 8e



6e4

+

65e2

+

34



sin E

−

1 48e



4e4

+

29e2

+

12



sin 3E

+

1 8



18e4

+

41e2

+

4



E

+

1 4



5e4

+

9e2

+

1



sin 2E

EF E0 p1

=

Γ

6 2 θF



θ0 cos

τ

sin

τ

(

1

+

e cos

τ

)

6d

τ

=

Γ

6 2

(

1

−

e2

₎

5



1 80e 3_{cos 5E}

₋

1 32



e2

+

3



e2cos 4E

+

7 8e



e2

+

2



cos E

+

1 16e



5e2

+

4



cos 3E

−

1 8



e4

+

9e2

+

2



cos 2E

EF E0 q1

=

Γ

6 2 θF



θ0 sin2

τ

(

1

+

e cos

τ

)

6d

τ

=

Γ

6 2

(

1

−

e2

₎

9/2



1 80e 3_{sin 5E}

₋

3 32e 2_{sin 4E}

₋

1 8e



e2

+

6



sin E

+

1 48e



e2

+

12



sin 3E

+

1 8



3e2

+

4



E

−

1 4sin 2E

EF E0

where E is the eccentric anomaly corresponding to the true anomaly

θ

, and the followings hold true[13]:

cos

θ

=

cos E

−

e 1

−

e cos E

,

sin

θ

=

sin E

√

1

−

e2 1

−

e cos E tan

θ

2

=



1

+

e 1

−

etan E 2

The last identity is very convenient because it alleviates the quad-rant ambiguity. From Eq. (27), we have convenient closed-form parameters to represent the Lagrange multipliers

λ

0 and

λ

1:



λ

0

λ

1

=

1 p0q1

−

p21



q1

−

p1

−

p1 p0



I0 I1

(28)

It is noted that p0, p1, q1, I0, and I1 are all constants. Finally, we use these parameters to express the optimal thrust acceleration Tz

(θ )

, producing: Tz

(θ )

=

az0 2

+

∞



n=1 azncos



2n

π

θ

θ



+



∞ n=1 bznsin



2n

π

θ

θ



=

1 2

θ

(λ

0fz0

+ λ1

gz0

)

+

1

θ

∞



n=1

λ

0fza

(

n

)

+ λ1

gza

(

n

)



cos



2n

π

θ

θ



+

1

θ

∞



n=1

λ

0fzb

(

n

)

+ λ1

gzb

(

n

)



sin



2n

π

θ

θ



(29)

Eq.(29)can be simplified into the following closed-form solution by Parseval’s theorem: Tz

(θ )

=

Γ

3 2



λ

0cos

θ

+ λ1

sin

θ

(

1

+

e cos

θ )

3

(30)

Eq. (30) is the final result for Tz

(θ )

which is a z-component of

thrust for the optimal reconfiguration of satellites in an elliptic or-bit. Furthermore, it is not difficult to show that Eq.(30)replicates Palmer’s result (Eq. (30) in[8]) when e

=

0. (SeeAppendix C.) We use Eq. (28) to evaluate

λ

0 and

λ

1 in which I0 and I1 are de-termined from Eq. (22). The performance index Eq.(21a) is then represented in a simple closed form as follows:

Jz

=

1

2

(

I0

λ

0

+

I1

λ

1

)

(31)

In addition, if Eq.(30)is inserted into Eqs.(11a) and (11b), we can obtain the variations in z and

˙

z:

˜

zp

(θ )

=

Γ

6 2

λ

0



sin

θ

A1

(θ )

−

cos

θ

A2

(θ )



+ λ1



sin

θ

A2

(θ )

−

cos

θ

A3

(θ )



˜

z_p

(θ )

=

Γ

6 2

λ

0



cos

θ

A1

(θ )

+

sin

θ

A2

(θ )



+ λ1



cos

θ

A2

(θ )

+

sin

θ

A3

(θ )



(32) where A1

(θ )

=

S



cos2

_θ

ρ

(θ )

6

=

θ



θ0 cos2

_τ

(

1

+

e cos

τ

)

6d

τ

A2

(θ )

=

S



cos

θ

sin

θ

ρ

(θ )

6

=

θ



θ0 cos

τ

sin

τ

(

1

+

e cos

τ

)

6d

τ

and A3

(θ )

=

S



sin2

θ

ρ

(θ )

6

=

θ



θ0 sin2

τ

(

1

+

e cos

τ

)

6d

τ

are given inAppendix B. From Eqs.(6a) and (6b), the actual posi-tion and velocity for the z-component are, respectively,

zp

(θ )

=

Γ

7 2

ρ

(θ )

λ

0



sin

θ

A1

(θ )

−

cos

θ

A2

(θ )



+ λ1



sin

θ

A2

(θ )

−

cos

θ

A3

(θ )



˙

zp

(θ )

=

Γ

5 2

λ

0



(

e

+

cos

θ )

A1

(θ )

+

sin

θ

A2

(θ )



+ λ1



(

e

+

cos

θ )

A2

(θ )

+

sin

θ

A3

(θ )



(33)

If zp

(θ )

and

˙

zp

(θ )

are found, we then know the z-component of

the Deputy’s position and velocity during the maneuver by adding the homogeneous solutions of Eq.(10b).